Cohomology theories for algebraic varieties

在V. Voevodsky的开创性工作之后，通过完全拓扑方法处理代数变量成为可能。在这方面，所谓的广义上同调理论发挥了重要作用。这包括经典的代数k理论，但也有一个相当现代（和更普遍）的代数协点理论。这些理论及其上同运算的研究是一个引人入胜的课题。它在代数几何、二次型理论和其他领域的经典问题中有许多应用。例如：罗斯特度公式，平滑代数循环的问题，以及场的u不变量。这是一个快速发展的新领域，提供了许多有前途的研究方向。

After the ground breaking works of V. Voevodsky, it became possible to work with algebraic varieties by completely topological methods. An important role in this context is played by the so-called Generalized Cohomology Theories. This includes classical algebraic K-theory, but also a rather modern (and more universal) Algebraic Cobordism theory. The study of such theories and cohomological operations on them is a fascinating subject. It has many applications to the classical questions from algebraic geometry, quadratic form theory, and other areas. One can mention, for example: the Rost degree formula, the problem of smoothing algebraic cycles, and u-invariants of fields. This is a new and rapidly developing area that offers many promising directions of research.